Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach

TL;DR

This work develops a discrete WKB framework to study local fermion densities in inhomogeneous XX chains with smoothly varying hoppings and magnetic fields. By solving the three-term recurrence for single-particle states in the large-$N$ limit, the authors derive a universal density formula $ ho(x,\\varepsilon_F)=\frac{1}{\pi a}\,\arccos(-\\xi^*(x,\\varepsilon_F))$ that captures depletion and saturation across arbitrary fillings and profiles. The approach is validated across several model chains (homogeneous, Krawtchouk, rainbow, cosine, and asymmetric cosine), showing excellent agreement with numerics and offering a path toward analytic entanglement characterizations beyond standard field-theoretic methods. This method thus provides both practical density predictions and a scaffold for understanding entanglement suppression in nonuniform free-fermion systems, with potential extensions to correlation matrices and entropy calculations at general fillings. The results bridge discrete lattice models with continuum WKB intuition, enabling analytical treatment of inhomogeneous fermionic chains away from criticality.

Abstract

We introduce a novel analytical approach for studying free-fermion (XX) chains with smoothly varying, site-dependent hoppings and magnetic fields. Building on a discrete WKB-like approximation applied directly to the recurrence relation for the single-particle eigenfunctions, we derive a closed-form expression for the local fermion density profile as a function of the Fermi energy, which is valid for arbitrary fillings, hopping amplitudes and magnetic fields. This formula reproduces the depletion and saturation effects observed in previous studies of inhomogeneous free-fermion chains, and provides a theoretical framework to understand entanglement entropy suppression in these models. We demonstrate the accuracy of our asymptotic formula in several chains with different hopping and magnetic field profiles. Our findings are thus the first step towards an analytical treatment of entanglement in free-fermion chains beyond the reach of conventional field-theoretic techniques.

Figures (14)

• Figure 1: Eigenfunctions of the general cosine chain \ref{['eq:gen_cos_chain']} with $N=400$ spins and energies $\varepsilon_{\frac{N}{2}+i}$ for $i=-20,-19,\dots,19$ (from top to bottom and left to right).
• Figure 2: Plots of $B(x)\pm 2J(x)$ for $B(x)=2J(x)$ (left) and $B(x)=-2J(x)$ (right), for a continuum coupling $J(x)$ vanishing at the origin. In the former case there is a single saturation interval $[0,x_1(\varepsilon_F)]$ for Fermi energies $\varepsilon_F\in[0,\varepsilon_1]$, where $x_1(\varepsilon_F)$ is the smallest positive root of the equation $4J(x)=\varepsilon_F$. A second, disjoint saturation interval $[x_2(\varepsilon_F),x_3(\varepsilon_F)]$ appears when $\varepsilon_F\in(\varepsilon_1,\varepsilon_2)$, where $x_2(\varepsilon_F)$ and $x_3(\varepsilon_F)$ are the second and third smallest positive roots of the equation $4J(x)=\varepsilon_F$ . These two saturation regions merge into a single interval $[0,x_1(\varepsilon_F)]$ for energies above $\varepsilon_2$. The situation is similar for the case $B(x)=-2J(x)$ represented in the right panel, the saturation intervals becoming depletion ones.
• Figure 3: Plot of $B(x)+2J(x)$ and $B(x)-2J(x)$ (respectively in red and blue) for the Krawtchouk chain of unit length with $0<q\leqslant1/2$ (left) and $1/2\leqslant q\leqslant 1$ (right).
• Figure 4: Left column: local fermion density for the Krawtchouk chain with $q=1/4$, $N=400$, and fillings $\nu=1/8$, $1/2$, and $7/8$ (blue markers), compared to its asymptotic approximation \ref{['eq:rho:Kraw']} (red line). Right column: analogous plot of the corresponding single-particle eigenfunctions $\phi_n(N\nu)$ (in blue) compared to their WKB envelopes \ref{['eq:phi:env']} (red lines).
• Figure 5: Schematic plot of $\pm 2J(x)$ for the rainbow chain.

Theorems & Definitions (7)

• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Remark 5.1